Let $X$ be a metric space, and let $f:X\rightarrow X$ be a homeomorphism. The $\omega$ limit set of $x\in X$ denoted by $\omega(x,f)$ is the set of cluster points of the forward orbit $\{f^n(x)\}_{n\in \N}$ Hence, $y\in \omega(x,f)$ if and only if there is a strictly increasing sequence of natural numbers $\{n_k\}_{k\in \N}$ such that $f^{n_k}(x)\rightarrow y$ as $k\rightarrow\infty$

Another way to express this is $$\omega(x,f) = \bigcap_{n\in \N} \overline{\{f^k(x): k>n\}}.$$

The $\alpha$ limit set is defined in a similar fashion, but for the backward orbit; i.e. $\alpha(x,f)=\omega(x,f^{-1})$

Both sets are $f$ invariant, and if $X$ is compact, they are compact and nonempty.

If $\varphi:\R\times X\to X$ is a continuous flow, the definition is similar: $\omega(x,\varphi)$ consists of those elements $y$ of $X$ for which there exists a strictly increasing sequnece $\{t_n\}$ of real numbers such that $t_n\rightarrow \infty$ and $\varphi(x,t_n) \rightarrow y$ as $n\rightarrow\infty$ Similarly, $\alpha(x,\varphi)$ is the $\omega$ limit set of the reversed flow (i.e. $\psi(x,t) = \phi(x,-t)$ . Again, these sets are invariant and if $X$ is compact they are compact and nonempty. Furthermore, $$\omega(x,f) = \bigcap_{n\in \N}\overline{\{\varphi(x,t):t>n\}}.$$